Ideals of étale groupoid algebras with coefficients in a sheaf with applications to topological dynamics

  • Gilles Gonçalves de Castro

    Universidade Federal de Santa Catarina, Florianópolis, Brazil
  • Daniel Gonçalves

    Universidade Federal de Santa Catarina, Florianópolis, Brazil
  • Benjamin Steinberg

    The City College of New York, USA
Ideals of étale groupoid algebras with coefficients in a sheaf with applications to topological dynamics cover
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Abstract

We prove the Effros–Hahn conjecture for groupoid algebras with coefficients in a sheaf, obtaining as a consequence a description of the ideals in skew inverse semigroup rings. We also use the description of the ideals to characterize when the groupoid algebras with coefficients in a sheaf are von Neumann regular, primitive, semiprimitive, or simple. We apply our results to the topological dynamics of actions of inverse semigroups, describing the existence of dense orbits and minimality in terms of primitivity and simplicity, respectively, of the associated algebra. Moreover, we apply our results to the usual complex groupoid algebra of continuous functions with compact support, used to build the -algebra associated with a groupoid, and describe criteria for its simplicity.

Cite this article

Gilles Gonçalves de Castro, Daniel Gonçalves, Benjamin Steinberg, Ideals of étale groupoid algebras with coefficients in a sheaf with applications to topological dynamics. Doc. Math. 30 (2025), no. 6, pp. 1421–1459

DOI 10.4171/DM/1035